Abstract
Engineering applications often comprise materials with an anisotropic response, eventually including nearly-inextensible fibers and/or small volumetric deformations. These behaviours can be regarded as internal constraints in material response. Numerical simulations comprising “constrained” materials show an overstiff structural response, referred to as element locking. Implementations based on mixed variational methods can heal locking but several solutions adopted in the state-of-the-art are still non-optimal for anisotropic materials. This issue is here analysed by discussing a decomposition of anisotropic deformation modes based on kinematic and energy criteria. Low-order mixed finite element formulations for dealing with near-inextensibility and/or near-incompressibility constraints are also introduced, and analysed on novel numerical applications of engineering interest.
Professor Wriggers has been my Mentor during my postdoctoral training from 2015 until 2019. I had the privilege to collaborate with Him on several topics, like multiphysical computational modelling for biomedical applications, the Virtual Element Method for homogenization, and mixed formulations for anisotropic materials. Discussions with Professor Wriggers have always been inspiring. His leadership and example have helped me grow into my potential and I would not be where and who I am today without Him. I am looking forward to many more meetings, especially those gathering scientific discussions with a glass of nice red wine, possibly on Hamburg’s pier.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Introducing \(\mathbf{A} = \mathbf{a} \otimes \mathbf{a}\), \(\mathbf{B} = \mathbf{I}-\mathbf{A}\), and tensor products as in [7], the Walpole’s basis reads:
$$\begin{aligned}&\mathbb {B}_1= \mathbf{A} \otimes \mathbf{A} \, , \qquad \mathbb {B}_2= \frac{1}{2} \mathbf{B} \otimes \mathbf{B} \, , \qquad \mathbb {B}_3= \frac{\sqrt{2}}{2} \mathbf{A} \otimes \mathbf{B} \, , \qquad \mathbb {B}_4= \frac{\sqrt{2}}{2} \mathbf{B} \otimes \mathbf{A} , \\&\mathbb {B}_5= \frac{1}{2} \left( \mathbf{B} \overline{\otimes } \mathbf{B} +\mathbf{B} \underline{\otimes } \mathbf{B} -\mathbf{B} \otimes \mathbf{B} \right) \, , \qquad \mathbb {B}_6= \frac{1}{2} \left( \mathbf{A} \overline{\otimes } \mathbf{B} +\mathbf{A} \underline{\otimes } \mathbf{B} +\mathbf{B} \overline{\otimes } \mathbf{A} +\mathbf{B} \underline{\otimes } \mathbf{A} \right) . \end{aligned}$$.
- 2.
Material constants in Eq. (1) for \(\mathbb {C}_\mathrm{{TI}}\) are obtained by employing a micromechanical approach which considers two isotropic constituents, i.e. fibers and matrix (see [7]). Fibers are significantly stiffer than the surrounding matrix (i.e., Young’s moduli \(E_f=10^{6}E_m\) with \(E_m=0.01\)), leading to near-inextensibility. Fiber volume fraction is chosen equal to \(V_f=10\%\), fiber Poisson’s ratio \(\nu _f=0.3\), while matrix Poisson’s ratio \(\nu _m\) is defined differently in the two addressed applications.
References
Zienkiewicz, O.C., Taylor, R.L., & Zhu, J.Z. (2013). The finite element method: Its basis and fundamentals. 7th ed. Butterworth-Heinemann, Oxford.
Wriggers, P. (2008). Nonlinear finite element methods. Berlin, Heidelberg: Springer.
Wriggers, P., Schröder, J., & Auricchio, F. (2016). Finite element formulations for large strain anisotropic material with inextensible fibers. Advanced Modeling and Simulation in Engineering Sciences, 3(1), 25.
Rasolofoson, F., Grieshaber, B. J., & Reddy, B. D. (2019). Finite element approximations for near-incompressible and near-inextensible transversely isotropic bodies. International Journal for Numerical Methods in Engineering, 117(6), 693–712.
Zdunek, A. (2020). On purely mechanical simple kinematic internal constraints. Journal of Elasticity, 139(1), 123–152.
Auricchio, F., Beirão da Veiga, L., Brezzi, F., & Lovadina, C. (2017) Mixed finite element methods. In Encyclopedia of computational mechanics (2nd ed., pp. 1–53). Wiley.
Marino, M., & Wriggers, P. (2020). Nearly-constrained transversely isotropic linear elasticity: Energetically consistent anisotropic deformation modes for mixed finite element formulations. International Journal of Solids and Structures, 202, 166–183.
Walpole, L. J. (1981). Elastic behavior of composite materials: Theoretical foundations. Advances in Applied Mechanics, 21, 169–242.
Hamila, N., & Boisse, P. (2013). Locking in simulation of composite reinforcement deformations. Analysis and treatment. Composites Part A: Applied Science and Manufacturing, 53, 109–117.
Potter, K. (2002). Bias extension measurements on cross-plied unidirectional prepreg. Composites Part A: Applied Science and Manufacturing, 33(1), 63–73.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Marino, M. (2022). It’s Too Stiff: On a Novel Mixed Finite Element Formulation for Nearly-Inextensible Materials. In: Aldakheel, F., Hudobivnik, B., Soleimani, M., Wessels, H., Weißenfels, C., Marino, M. (eds) Current Trends and Open Problems in Computational Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-87312-7_31
Download citation
DOI: https://doi.org/10.1007/978-3-030-87312-7_31
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-87311-0
Online ISBN: 978-3-030-87312-7
eBook Packages: EngineeringEngineering (R0)